Creating Symmetry: The Artful Mathematics of Wallpaper Patterns
by Frank A. Farris
Princeton University Press | June 2015 | ISBN-10: 0691161739 | PDF | 248 pages

This lavishly illustrated book provides a hands-on, step-by-step introduction to the intriguing mathematics of symmetry. Instead of breaking up patterns into blocks--a sort of potato-stamp method--Frank Farris offers a completely new waveform approach that enables you to create an endless variety of rosettes, friezes, and wallpaper patterns: dazzling art images where the beauty of nature meets the precision of mathematics.
Featuring more than 100 stunning color illustrations and requiring only a modest background in math, Creating Symmetry begins by addressing the enigma of a simple curve, whose curious symmetry seems unexplained by its formula.
Fun, accessible, and challenging, Creating Symmetry features numerous examples and exercises throughout, as well as engaging discussions of the history behind the mathematics presented in the book.

About the Author
Frank A. Farris teaches mathematics at Santa Clara University. He is a former editor of Mathematics Magazine, a publication of the Mathematical Association of America. He lives in San Jose, California.

CONTENTS
Preface vii
1 Going in Circles 1
2 Complex Numbers and Rotations 5
3 Symmetry of the Mystery Curve 11
4 Mathematical Structures and Symmetry: Groups, Vector Spaces, and More 17
5 Fourier Series: Superpositions of Waves 24
6 Beyond Curves: Plane Functions 34
7 Rosettes as Plane Functions 40
8 Frieze Functions (from Rosettes!) 50
9 Making Waves 60
10 Plane Wave Packets for 3-Fold Symmetry 66
11 Waves, Mirrors, and 3-Fold Symmetry 74
12 Wallpaper Groups and 3-Fold Symmetry 81
13 Forbidden Wallpaper Symmetry: 5-Fold Rotation 88
14 Beyond 3-Fold Symmetry: Lattices, Dual Lattices, and Waves 93
15 Wallpaper with a Square Lattice 97
16 Wallpaper with a Rhombic Lattice 104
17 Wallpaper with a Generic Lattice 109
18 Wallpaper with a Rectangular Lattice 112
19 Color-Reversing Wallpaper Functions 120
20 Color-Turning Wallpaper Functions 131
21 The Point Group and Counting the 17 141
22 Local Symmetry in Wallpaper and Rings of Integers 157
23 More about Friezes 168
24 Polyhedral Symmetry (in the Plane?) 172
25 Hyperbolic Wallpaper 189
26 Morphing Friezes and Mathematical Art 200
27 Epilog 206
A Cell Diagrams for the 17 Wallpaper Groups 209
B Recipes for Wallpaper Functions 211
C The 46 Color-Reversing Wallpaper Types 215
Bibliography 227
Index 229
 
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